Law Of Sines Worksheet Answers: Quick & Easy Solutions
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The Law of Sines is a fundamental concept in trigonometry, and it plays a critical role in solving problems involving triangles, especially when we lack certain measurements. In this article, we will explore the Law of Sines, provide some quick solutions to common problems, and present a worksheet that can help you practice and solidify your understanding of this essential topic.
Understanding the Law of Sines
The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. This can be expressed mathematically as:
$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $
Where:
- (a, b, c) are the lengths of the sides of the triangle,
- (A, B, C) are the angles opposite those sides.
When to Use the Law of Sines
The Law of Sines is particularly useful in the following scenarios:
- Angle-Angle-Side (AAS): Two angles and one non-included side are known.
- Angle-Side-Side (ASS): Two sides and a non-included angle are known. This case can lead to the ambiguous case, where two different triangles can be formed.
- Side-Side-Side (SSS): All three sides are known, allowing the calculation of all angles.
Example Problems Using the Law of Sines
Let's solve some example problems using the Law of Sines. These examples will also serve as answers to the worksheet you may encounter.
Example 1: Solving for a Side
Problem: In triangle ABC, (A = 30^\circ), (B = 60^\circ), and (a = 10). Find side (b).
Solution:
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Calculate angle C: $C = 180^\circ - A - B = 180^\circ - 30^\circ - 60^\circ = 90^\circ$
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Apply the Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B}$ $\frac{10}{\sin 30^\circ} = \frac{b}{\sin 60^\circ}$ $\frac{10}{0.5} = \frac{b}{\sqrt{3}/2}$
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Cross-multiply and solve for (b): $20 = \frac{b \cdot \sqrt{3}}{2}$ $b = \frac{20 \cdot 2}{\sqrt{3}} = \frac{40}{\sqrt{3}} \approx 23.09$
Example 2: Solving for an Angle
Problem: In triangle ABC, (a = 7), (b = 10), and (C = 45^\circ). Find angle (A).
Solution:
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Apply the Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B}$ $\frac{7}{\sin A} = \frac{10}{\sin 45^\circ}$ $\frac{7}{\sin A} = \frac{10}{\frac{\sqrt{2}}{2}}$
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Solve for ( \sin A ): $\sin A = \frac{7 \cdot \frac{\sqrt{2}}{2}}{10}$ $\sin A = \frac{7\sqrt{2}}{20}$
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Calculate (A): $A \approx \sin^{-1}(\frac{7\sqrt{2}}{20})$
Example 3: The Ambiguous Case (ASS)
Problem: Given triangle ABC, (a = 8), (b = 10), and (A = 30^\circ). Determine the possible values for angle (B).
Solution:
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Use the Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B}$ $\frac{8}{\sin 30^\circ} = \frac{10}{\sin B}$ $\frac{8}{0.5} = \frac{10}{\sin B}$ $16 = \frac{10}{\sin B}$ $\sin B = \frac{10}{16} = 0.625$
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Determine angles (B): $B = \sin^{-1}(0.625) \approx 38.68^\circ$ Also, consider the possibility: $B' = 180^\circ - 38.68^\circ \approx 141.32^\circ$
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Check if both are valid in triangle: For (B = 38.68^\circ), (C = 180^\circ - 30^\circ - 38.68^\circ = 111.32^\circ) (valid). For (B' = 141.32^\circ), (C' = 180^\circ - 30^\circ - 141.32^\circ = 8.68^\circ) (valid).
Thus, the possible angles for (B) are (38.68^\circ) and (141.32^\circ).
Practice Worksheet
Here's a basic worksheet to practice the Law of Sines:
<table> <tr> <th>Problem</th> <th>Find</th> </tr> <tr> <td>1. (A = 45^\circ), (B = 30^\circ), (a = 15)</td> <td>Find (b)</td> </tr> <tr> <td>2. (a = 12), (b = 15), (C = 50^\circ)</td> <td>Find angle (A)</td> </tr> <tr> <td>3. (a = 9), (b = 12), (A = 50^\circ)</td> <td>Determine angle (B)</td> </tr> </table>
Key Takeaways
- The Law of Sines is a powerful tool for solving triangles.
- It's essential to identify the type of problem before applying the law.
- Practice makes perfect! Keep working through examples to strengthen your understanding.
By mastering the Law of Sines, you can tackle a variety of problems confidently, whether in a classroom setting or real-world applications. Happy solving! 🎉